128
12 The Measurement of the Critical Velocity
and the same numerical value for this velocity c o in all reference systems
if we
synchronise the clocks in the moving systems according to our elementary principle
of relativity:
Two observers moving towards each other with a uniform velocity v measure one and the
same critical velocity c o .
This is the core of the Special Theory of Relativity.
Expressed differently: The result of a measurement of the critical velocity c o does
not depend on whether we move towards or away from the signal emitter, with the
velocity v. The equivalence to Special Relativity for our physical spacetime with
the speed of light c L is thus perfect. For our internal observers inside of our crystal,
the critical velocity c o (or the transversal sound velocity c T as we state with certain
justification) is an ‘absolute natural constant’. We can thus repeat this by saying:
The critical velocity c o of the sine-Gordon equation is, for the internal observers inside of
a crystal, an ‘absolute natural constant’.
This is nothing else than Einstein’s [14, 15] postulate, which we quoted on p. 12:
‘Any ray of light moves in the ‘stationary’ system of coordinates with a determined
velocity c whether the ray be emitted by a stationary or a moving body. Hence
velocity =
light path
time interval
where time interval is to be taken in the sense of the definition in Sect. 1’.
Here, we wish to note: We discovered the relativity of simultaneity before the
universal constancy of the critical velocity c o . This was based on our primary knowledge of the behaviour of our measuring-rods and clocks and with the help of the
synchronisation regulations according to our elementary principle of relativity. Once
the universal constancy of c o has been accepted, we can show the relativity of simultaneity for the internal observers just as we can show it according to Einstein [14,
15] using light.
Our observer in
observes that two c o -signals emitted from both end points of
our measuring section towards the central point were emitted at exactly the same
point of time, if they meet each other exactly in the centre of the measuring section.
For an observer in o , the signal coming from the left, in other words moving in
the direction of motion, needs according to Eq. (131) the time t → =
x/2
c o −v
to reach
the center of our measuring section x/2, whereas the signal coming from the right
needs only the time t ← =
x/2
c o +v
. The signal coming from the right must thus be
emitted at a later point of time than the signal coming from the left, if they are
to meet in the central point. According to the observer in o , the signals are thus
not emitted simultaneously. The reference systems o and
show themselves as
complete equivalent. Observed from o , we can state:
The coordinates x and t are the coefficients of measure of space and time measurement in
the reference system (x , t ), with the units of measure L and T .
12 The Measurement of the Critical Velocity
and the same numerical value for this velocity c o in all reference systems
if we
synchronise the clocks in the moving systems according to our elementary principle
of relativity:
Two observers moving towards each other with a uniform velocity v measure one and the
same critical velocity c o .
This is the core of the Special Theory of Relativity.
Expressed differently: The result of a measurement of the critical velocity c o does
not depend on whether we move towards or away from the signal emitter, with the
velocity v. The equivalence to Special Relativity for our physical spacetime with
the speed of light c L is thus perfect. For our internal observers inside of our crystal,
the critical velocity c o (or the transversal sound velocity c T as we state with certain
justification) is an ‘absolute natural constant’. We can thus repeat this by saying:
The critical velocity c o of the sine-Gordon equation is, for the internal observers inside of
a crystal, an ‘absolute natural constant’.
This is nothing else than Einstein’s [14, 15] postulate, which we quoted on p. 12:
‘Any ray of light moves in the ‘stationary’ system of coordinates with a determined
velocity c whether the ray be emitted by a stationary or a moving body. Hence
velocity =
light path
time interval
where time interval is to be taken in the sense of the definition in Sect. 1’.
Here, we wish to note: We discovered the relativity of simultaneity before the
universal constancy of the critical velocity c o . This was based on our primary knowledge of the behaviour of our measuring-rods and clocks and with the help of the
synchronisation regulations according to our elementary principle of relativity. Once
the universal constancy of c o has been accepted, we can show the relativity of simultaneity for the internal observers just as we can show it according to Einstein [14,
15] using light.
Our observer in
observes that two c o -signals emitted from both end points of
our measuring section towards the central point were emitted at exactly the same
point of time, if they meet each other exactly in the centre of the measuring section.
For an observer in o , the signal coming from the left, in other words moving in
the direction of motion, needs according to Eq. (131) the time t → =
x/2
c o −v
to reach
the center of our measuring section x/2, whereas the signal coming from the right
needs only the time t ← =
x/2
c o +v
. The signal coming from the right must thus be
emitted at a later point of time than the signal coming from the left, if they are
to meet in the central point. According to the observer in o , the signals are thus
not emitted simultaneously. The reference systems o and
show themselves as
complete equivalent. Observed from o , we can state:
The coordinates x and t are the coefficients of measure of space and time measurement in
the reference system (x , t ), with the units of measure L and T .